## Linear Operators: General theory |

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Page 24

Nelson Dunford, Jacob T. Schwartz. PROôF . If x € X , there is a sequence of

points an € A with an + x . Since { am } is a Cauchy sequence , and f is

continuous , the sequence { f ( an ) } is also a Cauchy sequence . Since Y is

complete ...

Nelson Dunford, Jacob T. Schwartz. PROôF . If x € X , there is a sequence of

points an € A with an + x . Since { am } is a Cauchy sequence , and f is

**uniformly**continuous , the sequence { f ( an ) } is also a Cauchy sequence . Since Y is

complete ...

Page 145

A sequence of functions { In } defined on S with values in X converges u -

In } converges

the ...

A sequence of functions { In } defined on S with values in X converges u -

**uniformly**if for each ε > 0 there is a set EEE such that v ( u , E ) < € and such that {In } converges

**uniformly**on S— E. The sequence { In } converges u -**uniformly**tothe ...

Page 360

which vanishes in a neighborhood of a point p the sequence ( Sn ) ( x )

converges to zero

En ( x , x ) dz 5 M , then the convergence of Snf for a given c . o . n . system is

localized if ...

which vanishes in a neighborhood of a point p the sequence ( Sn ) ( x )

converges to zero

**uniformly**for x in some neighborhood of p . 21 Show that if SEn ( x , x ) dz 5 M , then the convergence of Snf for a given c . o . n . system is

localized if ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero